The cubic scroll in P^4 is the blowup of the projective plane at a single point. Here this surface is projected to P^3 from a point not on it. The result is a surface with a double line (the image of the directrix) which is a ruled cubic. Classically this surface is called the Plucker conchoid , given by affine equations y^z= (1-z)x^2, or parametrically by (t^2+u^2,st,su,t^2) in P^3.